On Boolean posets of numerical events.

With many physical processes in which quantum mechanical phenomena can occur, it is essential to take into account a decision mechanism based on measurement data. This can be achieved by means of so-called numerical events, which are specified as follows: Let be a set of states of a physical system and () the probability of the occurrence of an event when the system is in state . A function is called a numerical event or alternatively, an -probability. If a set of -probabilities is ordered by the order of real functions, it becomes a poset which can be considered as a quantum logic. In case the logic is a Boolean algebra, this will indicate that the underlying physical system is a classical one. The goal of this paper is to study sets of -probabilities which are not far from being Boolean algebras by means of the addition and comparison of functions that occur in these sets. In particular, certain classes of so-called Boolean posets of -probabilities are characterized and related to each other and descriptions based on sets of states are derived.